Simplify and expand the following expression: $ \dfrac{4}{4p - 4}- \dfrac{2}{p + 10}- \dfrac{1}{p^2 + 9p - 10} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $4$ out of denominator in the first term: $ \dfrac{4}{4p - 4} = \dfrac{4}{4(p - 1)}$ We can factor the quadratic in the third term: $ \dfrac{1}{p^2 + 9p - 10} = \dfrac{1}{(p - 1)(p + 10)}$ Now we have: $ \dfrac{4}{4(p - 1)}- \dfrac{2}{p + 10}- \dfrac{1}{(p - 1)(p + 10)} $ The least common multiple of the denominators is: $ 4(p - 1)(p + 10)$ In order to get the first term over $4(p - 1)(p + 10)$ , multiply by $\dfrac{p + 10}{p + 10}$ $ \dfrac{4}{4(p - 1)} \times \dfrac{p + 10}{p + 10} = \dfrac{4(p + 10)}{4(p - 1)(p + 10)} $ In order to get the second term over $4(p - 1)(p + 10)$ , multiply by $\dfrac{4(p - 1)}{4(p - 1)}$ $ \dfrac{2}{p + 10} \times \dfrac{4(p - 1)}{4(p - 1)} = \dfrac{8(p - 1)}{4(p - 1)(p + 10)} $ In order to get the third term over $4(p - 1)(p + 10)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{1}{(p - 1)(p + 10)} \times \dfrac{4}{4} = \dfrac{4}{4(p - 1)(p + 10)} $ Now we have: $ \dfrac{4(p + 10)}{4(p - 1)(p + 10)} - \dfrac{8(p - 1)}{4(p - 1)(p + 10)} - \dfrac{4}{4(p - 1)(p + 10)} $ $ = \dfrac{ 4(p + 10) - 8(p - 1) - 4} {4(p - 1)(p + 10)} $ Expand: $ = \dfrac{4p + 40 - 8p + 8 - 4}{4p^2 + 36p - 40} $ $ = \dfrac{-4p + 44}{4p^2 + 36p - 40}$ Simplify: $ = \dfrac{-p + 11}{p^2 + 9p - 10}$